Chemical Erosion of Refractory-Metal Nozzle Inserts inSolid-Propellant Rocket Motors

Piyush Thakre∗ and Vigor Yang†

Pennsylvania State University,

University Park, Pennsylvania 16802

DOI: 10.2514/1.37922

An integrated theoretical/numerical framework is established and validated to study the chemical erosion of

refractory-metal (tungsten, rhenium, and molybdenum) nozzle inserts in solid-rocket-motor environments, with a

primary focus on tungsten. The formulation takes into account multicomponent thermofluid dynamics in the gas

phase, heterogeneous reactions at the surface, energy transport in the solid phase, and nozzle material properties.

Typical combustion species of nonmetallized ammonium-perchlorate/hydroxyl-terminated-polybutadiene

propellants at practical motor operating conditions are considered. The erosion rates calculated by employing

three different sets of chemical kinetics data available in the literature for the tungsten-steam reaction have been

compared. The effect of considering either of two different tungsten oxides, WO2 or WO3, as the final product of

surface reactions is also investigated. The predicted erosion rates compare well with experimental data. The

oxidizing species ofH2O proved more detrimental than CO2 in dictating the tungsten nozzle erosion. The material

recession rate is controlled by heterogeneous chemical kinetics because the diffusion limit is not reached. The erosion

rate increases with increasing chamber pressure, mainly due to higher convective heat transfer and enhanced

heterogeneous surface reactions. The tungsten nozzle erodes much more slowly than graphite, but at a rate

comparable with that of rhenium. The molybdenum nozzle exhibits the least erosion for flame temperatures lower

than 2860 K. Its low melting temperature (2896 K), however, restricts applications for propellants with high flame

temperatures.

Nomenclature

Ai = preexponential factor for the rate constant inreaction i

Dim = molecular mass diffusivityEi = activation energy for reaction iG = Gibbs free energyh, H = enthalpyk = turbulent kinetic energy_m = mass flow rateN = total species numbersp = pressurept = chamber pressureRe = Reynolds numberRu = universal gas constant_rc = net surface recession rate, m=s_ri;ch = chemical-kinetics-controlled recession rate due to

species i, kg=m2 � s_ri;diff- lim = diffusion-controlled recession rate due to species i,

kg=m2 � s_ri;erosion = recession rate due to species i, kg=m2 � sT = temperatureTt = chamber temperatureUk = mass diffusion velocity of species ku, v, w = x, y, and z components of velocityWk = molecular weight of species kWmix = average molecular weight of the gases

Wr = molecular weight of refractory metal_w = species molar production rateYk = mass fraction of species k� = thermal diffusivity�GR = change in Gibbs free energy�HR = heat of reaction�h�f = heat of formation (at 298.15 K)" = dissipation rate� = thermal conductivity� = density_! = species mass production rate!i;diff- lim = maximum diffusion rate of species i toward the

surface, kg=m2 � s

Subscripts

amb = ambient conditionsc = solid phase (refractory metal)c–g = gas–solid interfaceg = gas phaseo = outer boundary of nozzle materials = surface

I. Introduction

T HE erosion of rocket-nozzle materials during motor firingscontinues to be one of the major hindrances in the advancement

of solid-rocket propulsion. A noneroding nozzle is desired, as itmaintains a constant nozzle expansion ratio and optimal thrustperformance. Graphite and carbon–carbon composites, which havebeen widely used as nozzle materials, may undergo significanterosion at high chamber pressures and surface temperatures [1,2].Because a throat-area increase of more than 5% is consideredexcessive for most solid-rocket applications [3], graphite erosionlevels for ultrahigh pressures and long-duration firings can becomeunacceptable. Different types of erosion-resistant nozzle materialsare thus required to further the development of rocket technology. Interms of highmelting temperature, refractorymetals such as tungsten(W at 3695 K), rhenium (Re at 3459 K), tantalum (Ta at 3290 K),

Presented as Paper 1030 at the 46thAerospace SciencesMeeting of AIAA,Reno, NV, 7–11 January 2008; received 6 April 2008; revision received 24August 2008; accepted for publication 5 September 2008. Copyright © 2008by the authors. Published by the American Institute of Aeronautics andAstronautics, Inc., with permission. Copies of this paper may be made forpersonal or internal use, on condition that the copier pay the $10.00 per-copyfee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,MA 01923; include the code 0748-4658/09 $10.00 in correspondence withthe CCC.

∗Ph.D. Candidate, Department of Mechanical and Nuclear Engineering;pkt109@psu.edu. Member AIAA.

†John L. and Genevieve H. McCain Chair in Engineering, Department ofMechanical and Nuclear Engineering; vigor@psu.edu. Fellow AIAA.

JOURNAL OF PROPULSION AND POWER

Vol. 25, No. 1, January–February 2009

40

molybdenum (Mo at 2896 K), and niobium (Nb at 2750 K) are veryattractive. The flame temperatures for aluminized ammonium-perchlorate/hydroxyl-terminated-polybutadiene (AP/HTPB) pro-pellants may reach 3700 K, making tungsten and rhenium promisingcandidates. One needs to consider, however, not only the meltingpoints of these pure metals, but also their oxides, carbides, andnitrides, because such compounds/eutectics can be formed at thenozzle surface and may modify the chemical behavior of the nozzlematerial. In general, tungsten nitrides are unstable and dissociatebelow 1100 K [4]. Consequently, oxidation is likely to dominate,with nitrogen serving as a diluent.

Refractory metals (W, Mo, and Re) and their alloys and carbides[e.g., zirconium carbide (ZrC), tantalum carbide (TaC), and hafniumcarbide (HfC)] have exhibited a potential to provide zero-erosionperformance [5,6], due to their resistance to chemical attack at hightemperatures. Thematerial and processing costs associatedwith thesemetals, however, are relatively high. In addition, the high materialdensity increases the inert mass of the vehicle. Metal carbides evensuffer from poor thermal shock resistance, leading to the formation ofcracks [5,7]. Refractory metals may also form interstitial solutionswith the interdiffusion of O, N, or C, which will make the materialsmore brittle and susceptible to cracking and spalling.

A fundamental understanding of the physiochemical processesdictating the erosion of nozzle material is critical to the developmentof new materials that can resist erosion more effectively.Experimental data with different materials are scarce, not onlybecause of the high costs involved in conducting measurements atpractical rocket-motor conditions, but also because of the proprietarynature of the data. Only a limited amount of information is availablein the open literature [5,7,8]. Olcott and Batchelor [5] studiedtungsten nozzle inserts and attributed the two principal failuremechanisms to the chemical attack of the propellant combustiongases, H2O and CO2, and cracking due to thermal stresses. Resultsfrom their subscale motor firings indicated a higher level of erosionwith propellants possessing a higher oxidation potential. Theaddition of 2% thorium dioxide (ThO2) helped reduce the erosionrate. Arc-cast tungsten with coarse grain was found to be moreresistant to the thermal-stress induced fracture than fine-graintungsten.

Johnston et al. [7] experimentally investigated the performance ofa variety of rocket-nozzle materials, including refractory metals (Wand Mo), metal carbides (ZrC, TaC, and HfC), graphite, and fiber-reinforced plastics. Three different propellants with flame temper-atures of 2865, 3365, and 3810 K were used. Some of the metal-carbide nozzles showed outstanding erosion resistance, comparablewith that of the best refractory-metal nozzles. However, all of themetal-carbide nozzles cracked as a result of thermal stresses.Haugen‡ [8] tested different nozzle materials (including W, Mo,Mo=Re, and W coated with Re) with nonmetallized AP/HTPBpropellants.

The overall nozzle erosion process is extremely complex,comprising the interplay of numerous factors, including propellantcomposition, motor operating conditions, nozzle geometry andmaterial properties, transport of reacting species, and chemicalreactions in the gas phase and at the nozzle surface. In addition tochemical erosion, there may also be erosion due to mechanicalprocesses associated with the impingement of metal-oxide particles(e.g., Al2O3�l�) onto the nozzle surface. It is imperative to develop acomprehensive model to accurately explore the erosion processunder various rocket-motor operating conditions and to identifymaterials with effective resistance to erosion. Although a fewmodels[9–12] have been established for treating graphite nozzle erosion,until now there have been few efforts to develop such models forrefractory metals. This paper presents the first comprehensiveanalysis for studying the chemical erosion of refractory-metal nozzleinserts under a broad range of practical rocket-motor environments.Various key mechanisms dictating the nozzle erosion rate for AP/HTPB composite propellants are identified and quantified.

II. Theoretical Formulation

The present work follows the general approach detailed in [12] forgraphite erosion and incorporates the physiochemical properties ofrefractory metals in the combustion streams of AP/HTPBpropellants. The formulation involves general conservationequations for the gas phase, energy transport in the solid phase,interfacial conditions between the gas and solid phases, and the outerboundary condition of the nozzle material. The gas-phase dynamicsare modeled using the Favre-averaged conservation equations ofmass, momentum, energy, and species concentration in axisym-metric coordinates. Turbulence closure is achieved by means of awell-calibrated two-layer turbulence model suitable for transpirationand accelerating flows [12–14]. Full account is taken of variabletransport and thermodynamic properties [12].

The current work aims to study the erosion of three refractorymetals (W, Re, and Mo), with a primary focus on tungsten. Mostexisting studies [7,8] have concluded that chemical erosion is theprincipal cause for the surface recession in nonmetallized propellantenvironments. In the case of aluminized propellants, mechanicalerosion caused by the impingement of metal-oxide particles couldalso be significant. In addition, there is the possibility of thermitereactions between unoxidized aluminum and metal oxides (e.g.,2Al�WO3 ! Al2O3 �W or 2Al�MoO3 ! Al2O3 �Mo)[15]. Such highly exothermic reactionsmay increase the temperatureto a level sufficient tomelt the metal nozzle surface [16]. The wettingof metal surfaces by Al2O3�l� droplets, on the other hand, provides aliquid coating that may help resist further oxidation and erosion. Thedeposition of Al2O3�l� may also increase the heat-transfer rate to thenozzle wall. All of the phenomena related to aluminized propellantsare still relatively poorly understood and require furtherinvestigation. To avoid such complications, only chemical erosionassociated with nonmetallized AP/HTPB solid propellants isconsidered. At sufficiently high temperatures, the metal surface isprone to chemical attack by such oxidizing species asH2O,CO2,O2,and O. Although the reactivity of metals with oxygen is significant[17,18], this effect can be ignored due to the fuel-rich nature of AP/HTPB propellants, which produce negligible amounts ofO2 andO inthe gas phase. Themain species considered in the combustion streamare thus H2O, CO2, CO, HCl, N2, and H2. Minor species withnegligible concentrations (such as NO, OH, H, O2 and O) areignored. The present study further neglects changes in thephysiochemical properties of nozzle materials and variations of theirmelting temperatures due to the possibility of the formation of anyeutectics.

A. Heterogeneous Kinetics of Tungsten

Tungsten (W)material is prone to chemical attack by the oxidizingspecies of H2O and CO2 present in the combustion stream of AP/HTPB solid propellant. The effect of HCl is negligible, according tothe experimental study by Farber [19]. Chemical-equilibriumcalculations [20] also suggest that HCl, H2, N2, and CO do not reactwith W, whereas H2O and CO2 show significant reactions. Olcottand Batchelor [5], however, observed that the presence of H2 andCO, respectively, reduces the reaction rates ofH2O andCO2 withW.A similar reduction of CO2 reaction with W due to the presence ofCO was observed by Walsh et al. [21].

Several experimental studies have been devoted to the oxidation oftungsten with steam [5,22–26] and CO2 [5,21] at high temperatures.There seems to be a fair bit of disagreement about the formof thefinaloxidation product. Some consider [5,23] WO3�g� as the majoroxidation product, whereas others [21,27] have suggestedWO2�g� asthe major product. Greene and Finfork [25] found that the oxideformed has a stoichiometry ofWO2:7, which indicates that bothWO3

and WO2 may be present. Kalipatrick and Lott [24] performedexperiments over a temperature range of 1273–1973 K to determinethe mechanism of the tungsten reaction with steam at atmosphericpressure. The measured activation energies were 48.9 and22:7 kcal=mol in the temperature ranges of 1273–1723 and 1723–1973 K, respectively. At temperatures greater than 1723 K, it wassuggested that the final oxidation products could be either the vapor‡Private communication with S. Haugen, 2007.

THAKRE AND YANG 41

phaseWO3 and its polymers or the volatileWO3 � H2O�g�. The rate-determining step was proposed to be the oxidation of WO2�s� toWO3 � H2O�g� and WO3�g� and its polymers. Based on theobservations of Belton and McCarron [26], the major species couldbe WO3 � H2O�g�, which is formed due to the reaction of WO3 andH2O in the gas phase. Such a reaction, if it occurs, could be effectiveat reducing the amount of H2O attacking the surface and hence theerosion rate. The chemical-equilibrium calculations [20] also showthat WO3�l;g� and its polymers are the major products at hightemperature, as described inTable 1. For temperatures above 1700K,a thermodynamic analysis based on Gibbs free energy indicates thatformation of WO3�g� is favored (Appendix A). In the current study,both WO3�g� and WO2�g� are considered individually as the finaloxidation product. The chemical pathways and kinetics data aresummarized in Table 2. The decrease in the reaction rate of CO2 andWdue the presence ofCO is accounted as a function of the ratio of themole fractions of CO and CO2 [21]. Whenever the reaction orderwith respect to H2O is not mentioned explicitly in the literature, afirst-order reaction is assumed.

B. Heterogeneous Kinetics of Molybdenum

Although molybdenum (Mo) has performed satisfactorily as anozzle throat insert and a structural support material, its usefulness islimited by its relatively high ductile-to-brittle transition temperature[3] and low melting point. The use of Mo inserts is usually restrictedto propellants with lower flame temperatures. The oxidation ofmolybdenum at high temperature yields H2�g� and volatile MoO3

[28,29]. The chemical-equilibrium calculations [20] show thatMoO3�g� and its polymers are the major products at hightemperatures, as listed in Table 3. Kalipatrick and Lott [29]experimentally investigated the oxidation of Mo under differentmass flow rates of steam. In the current study, the kinetics datacorresponding to the maximum mass flow rate are adopted, as theyrepresent the condition closest to the kinetically limited reactionrates. Table 2 gives the chemical kinetics data for the oxidation ofMoby steam at high temperatures. First-order kinetics with respect toH2O are assumed, as it is not explicitly mentioned in [29]. Thereaction ofCO2withMo is not included in the presentwork due to thelack of relevant kinetics data in the literature. The thermodynamic

analysis based on Gibbs free energy (Appendix A), however,indicates that the reaction of Mo with CO2 may be as significant asthat of W with CO2.

C. Heterogeneous Kinetics of Rhenium

Rhenium (Re)-based nozzle inserts can be used for propellantswith high flame temperatures due to the high melting temperature ofrhenium. They are also widely used in small liquid rocket engines[30]. The oxidation of rhenium by steam at high temperatures hasbeen experimentally studied [31,32]. Kalipatrick and Lott [31]suggested that the oxide formed is Re2O7, which is volatile at hightemperatures. Table 2 lists the chemical kinetics data for theoxidation of Re by steam. Duriez [32] concluded that the reaction isfirst order with respect to H2O. The reaction rate was found to behighly dependent on the mass flow rate of the steam and the gasvelocity, indicating that the experimental conditions were not closeto the kinetically limited case. In the current study, the kinetics data ofKalipatrick and Lott [31] are chosen because the data fit well for allthe steam flow rates considered. The reaction of CO2 with Re is notincluded, as no kinetics data relevant to this reaction is found in theliterature. Unlike the reaction ofCO2 andW, the reaction ofCO2 andRe is not likely to be significant, due to the higher oxidationresistance of Re. No thermodynamic analysis of the oxidationreaction based on Gibbs free energy was performed because of thelack of thermodynamic properties for rhenium oxides.

D. Chemical Reactions at the Nozzle Surface

The rate of consumption of metal by an oxidizing species i isexpressed as

_r i;ch � kipni;s �kg=m2 s� (1)

where

pi;s � psYi;sWmix;s

Wi

(2)

ki � Ai exp��Ei=RuTs� (3)

Table 1 Chemical-equilibrium calculations [20] for tungsten oxidation (reactants at 2800 K, 40 atm, and mass ratio of 1:1)

Reactants H2O CO2 WO3�l� WO3 and its polymers H2 CO WO2 �HR, kJ=kg �GR, kJ=kg

W=H2O 0.35 —— 0.12 0.51 0.02 —— �0 �2910:20 �23; 939:10W=CO2 —— 0.14 0.42 0.21 —— 0.23 �0 �2285:98 �13; 354:6

Table 2 Chemical kinetics data for heterogeneous surface reactions of tungsten

Surface reaction Ai Ei, kcal/mol _!, kg=m2 � s Temperature range, K Reference

Tungstena

W� 3H2O�g� !WO3�g� � 3H2�g� 8.64 kg=m2 � s � atm 22.70 kipH2O1723–1973 Kalipatrick and Lott [24]

or W� 2H2O�g� !WO2�g� � 2H2�g� 5:722 104 kg=m2 � s � atm 48.13 kipH2O1073–1973 Unal et al. [22]

W� 3CO2�g� !WO3�g� � 3CO�g� 1:433 104 kg=m2 � s � atm 47.23 kipH2O1073–1623 Greene and Finfork [25]

or W� 2CO2�g� !WO2�g� � 2CO�g� 4:026 103 79.00 kip0:88CO2

2200–3200 Walsh et al. [21]

Molybdenuma

Mo� 3H2O�g� ! MoO3�g� � 3H2�g� 4.48 kg=m2 � s � atm 57.5 kipH2O1373–1973 Kalipatrick and Lott [29]

Rheniuma

2Re� 7H2O�g� ! Re2O7�g� � 7H2�g� 29.85 kg=m2 � s � atm 29.80 kipH2O1123–1973 Kalipatrick and Lott [31]

2Re� 7H2O�g� ! Re2O7�g� � 7H2�g� 57.97 kg=m2 � s � atm 34.90 kipH2O1873–2473 Duriez [32]

aThe rate of consumption is obtained in kg=m2=s; ki � Ai exp��Ei=RuTs�.

Table 3 Chemical-equilibrium calculations [20] for molybdenum oxidation (reactants at 2800 K, 40 atm, and mass ratio 1:1)

Reactants H2O CO2 Mo�c� MoO3 and its polymers H2 CO �HR, kJ=kg �GR, kJ=kg

Mo=H2O 0.36 —— 0.26 0.36 0.015 —— �2497:07 �24; 292:3Mo=CO2 —— 0.15 0.25 0.38 —— 0.22 �1828:15 �13; 692:9

42 THAKRE AND YANG

In the preceding equations, Yi;s and pi;s represent the mass fractionand partial pressure of species i at the surface, respectively; Wmix;s,ps, and Ts are the molecular weight of the gas mixture, pressure, andtemperature at the surface, respectively; and n is the overall order ofthe heterogeneous reaction. The mass rate of consumption of anoxidizing species i at the gas–solid interface is given by [12]

�_! i � _ri;ch�iWi

�rWr

(4)

where �i and �r are the stoichiometric coefficients for the particularsurface reaction under consideration, as listed in Table 2.

E. Solid-Phase Governing Equation

With the neglect of thermal decomposition and chemical reactionsin the solid phase, the heat conduction in the radial direction isgoverned by the following equation:

�c@hc@t� �cr

@

@r�rhc _rc� �

1

r

@

@r

��cr

@Tc@r

�(5)

The equation takes into account the effect of surface recession andvariable thermophysical properties. The integration of Eq. (5) at asteady-state condition across the nozzle material gives

ri

��c@Tc@r

�ri

� _rc�c�hc–gri � horo� � ro��c@Tc@r

�ro

(6)

where ri and ro are the inner and outer radii, respectively, of thenozzle material at any axial location, and c–g and ho are thecorresponding specific enthalpies.

In most existing studies, the outer boundary of the nozzle materialis modeled to be adiabatic. This treatment is valid when the nozzlematerial is sufficiently thick or well insulated and the thermalconductivity is low. Considering the high thermal conductivity ofrefractory metals, however, the thermal penetration depth (��c= _rc)in the nozzlematerial under typical motor operating conditions couldbe of the same order as the thickness of the insert. Thus, theenforcement of the adiabatic condition needs to be carefullyexamined. An adequate sensitivity study on the effect of the outerboundary condition on the nozzlematerial erosion is elaborated later.

F. Gas–Solid Interfacial Condition

The processes in the gas and solid phases arematched at the nozzlesurface by enforcing the continuities of mass, species, and energyfluxes. This procedure eventually gives the erosion rate of nozzlematerial and the surface temperature. The conservation laws at thegas–solid interface can be written as follows:

Mass:

�� g ~ur � �c _rc (7)

Species:

�� ��gDkm

d ~Ykdr� ��g ~Yk ~ur

�� �_!k (8)

Energy:

��c@Tc@r

�ri

� _rc�chc–g ���g@ ~Tg@r

�ri

�XNk�1

�_!k ~hg;k (9)

where ~ur stands for the radial velocity in the gas phase due tomaterialerosion. The rate of production of the gas-phase species k at thenozzle surface on account of heterogeneous reactions is denoted by�_!k. The first term in Eq. (9) can be obtained by considering theoverall energy balance in the solid phase represented by Eq. (6).Radiation is neglected in Eq. (9) due to the prevalence of convectiveheat transfer in a nonmetallized propellant environment. Flowsymmetry is enforced along the nozzle centerline.

G. Nozzle Recession Rate

Owing to heterogeneous surface reactions, species-concentrationgradients are formed in the nozzleflowfield and cause the diffusion ofthose species toward or away from the surface. At a high surfacetemperature, heterogeneous chemical reactions can proceed sorapidly that the erosion rate could be dictated entirely by speciesdiffusion. The diffusion-controlled recession rate _ri;diff- lim due to aspecific oxidizing species i can be determined by first calculating�_!i;diff- lim from Eq. (8) with ~Yi � 0 and then applying the followingequation:

_r i;diff- lim � �_!i;diff- lim�rWr

�iWi

(10)

At low surface temperatures, heterogeneous reactions become therate-controlling process for nozzle erosion, due to reduced chemicalactivity. The recession rate (kg=m2 � s) associated with an oxidizingspecies i is obtained as

_r i;erosion �min� _ri;diff- lim; _ri;ch� (11)

The net recession rate (m=s) of the nozzle surface is determined by

_r c�x� �1

�c

Xi

_ri;erosion (12)

III. Numerical Treatment

The governing equations and associated boundary conditions aresolved numerically by means of a density-based finite volumeapproach with body-fitted coordinates. A four-stage Runge–Kuttascheme is used for the time integration. The convective fluxes aretreated explicitly with a second-order central-difference scheme,following the methodology proposed by Rai and Chakravarthy [33].The chemical reaction source terms are handled in a semi-implicitmanner. To ensure numerical stability and convergence, a fourth-order artificial dissipation based on the scalar dissipation model bySwanson and Turkel [34] is employed. The code has beenimplemented on a parallel computing facility by employing adistributed-memory message passing interconnection. The grid isstretched in the radial direction and clustered near the surface. Thecenters of the computational cells adjacent to the nozzle surface arelocated at y� < 1 to accurately capture the near-wall phenomena.

IV. Nozzle Configurations and Boundary Conditions

Figure 1 shows the baseline nozzle configuration considered here.The geometry is identical to that employed in our previous study ongraphite material [12] so that a direct comparison of the erosion ratesof different materials can bemade. The incoming flow consists of thecombustion products of nonmetallized AP/HTPB compositepropellants. The chamber pressure pt and temperature Tt arespecified at the nozzle inlet. The velocity at the exit is supersonic.Table 4 lists the species mass fractions at the inlet obtained from thechemical-equilibrium calculation [20] at pt � 6:9 MPa. Sixdifferent chamber pressures and their corresponding temperaturesare used to study the effect of motor operating conditions on nozzleerosion. The species mass fractions remain nearly constant in thepressure range of 6.9–45 MPa. The ambient temperature is taken as300 K. The thermophysical properties ofW,Mo, and Re are adoptedas polynomial functions of temperature.

The discussion of results is organized as follows. First, tungstenerosion for the baseline nozzle configuration is studied by employingthe different surface chemical kinetic schemes outlined in Sec. II.Based on the results, appropriate kinetics data and reaction productsare chosen for further parametric studies. The chemical erosionmodel is then validated against experimental data forW,Mo, andRe,as summarized in Table 5. The nozzle geometry and inlet flowconditions exactly simulate those in the corresponding experimentalstudies.

THAKRE AND YANG 43

V. Results and Discussions

The theoretical/numerical framework described in the precedingsections was implemented to explore the chemical erosion ofrefractory-metal (W, Mo, and Re) nozzles in practical rocket-motorenvironments. The axisymmetric computational domain of therocket nozzle, shown in Fig. 1, is divided into 141 80 grid points inthe x and r directions. The turbulent flow development in the samenozzle configuration was detailed earlier [12]. To simulate theerosion of the tungsten nozzle, two heterogeneous reactions of Wwith H2O and CO2, along with the energy balance given by Eq. (9),are implemented at the surface. The oxidizing species CO2 andH2Oare consumed to formCO,H2, and gaseous tungsten oxides (WO3 orWO2). Concentration gradients are then established near the wall.The gas-phase reactions among the AP/HTPB combustion-productspecies were not included because of their negligible effect on nozzleerosion [12]. It should be noted that for tungsten and other refractorymetals, volatile oxides (e.g., WO3 and WO2) formed at the nozzlesurface may react with HCl�g� to generate compounds such asWO2Cl2�g�. Such gas-phase reactions do not have any significantimpact on the nozzle erosion rate, as they will not affect either theconcentrations ofH2O and CO2 species or the rate of heat transfer tothe nozzle wall.

As shown inTable 2, three different sets of kinetics data [22,24,25]are available for the reaction between W and H2O at hightemperatures. It is important to make a judicious selection of the datathat yield the most accurate erosion rate. Figure 2 shows acomparison of the predicted tungsten nozzle erosion rates based ondifferent chemical kinetics data. The incoming flow temperature andpressure are Tt � 3000 K and pt � 6:9 MPa, respectively, and theouter boundary of the nozzle material is assumed to be adiabatic. Themaximum erosion rates obtained from the three sets of kinetics data[22,24,25] are 0.078, 0.069, and 0:047 mm=s, respectively. Themost severe erosion occurs near the throat, due to the enhanced heattransfer in that region. Themeasured erosion rates of tungsten nozzleinserts obtained fromvarious experimental studies§ [7,8]with similarnozzle throat diameters consistently lie in the range of0:02–0:05 mm=s. On this basis, the kinetics data of Kalipatrickand Lott [24], which lead to the lowest erosion rate, appear to bemostappropriate for the present application. The small wiggle in theerosion-rate profile arises from the discontinuity in the slope ofnozzle contour at the entrance of the throat region. Such smallirregularities were not observed in other cases with smooth-nozzleprofiles shown later.

The stoichiometry of surface reactions and the overall heat ofreaction depend strongly on the final product species, which in turnaffect the calculated surface temperature and erosion rate. Existingexperimental studies on tungsten oxidation [5,21,23,27] havesuggested eitherWO2�g� orWO3�g� as the final oxidation product. Toclarify the effect of this choice on the predicted erosion rates,calculations were performed for both WO2�g� or WO3�g�. Figure 3shows a comparison of the predicted tungsten nozzle erosion ratesobtained by considering either WO3�g� (�h

of ��319:725 kJ=mol)

orWO2�g� (�hof � 29:062 kJ=mol) as the final oxidation product of

surface reactions. The chemical kinetics data from Kalipatrick and

Lott [24] are used. The use ofWO3�g� gives rise to an erosion rate thatis 0:005 mm=s lower. It is possible that if the oxidation productemerging from the surface isWO2, then itmay further reactwithH2Oto form eitherWO3�g� orWO2�OH�2�g� [24]. This process reduces theoxidation potential of combustion products by depleting theconcentration of H2O, thereby causing a reduced erosion rate. Asimilar situation occurs if the oxidation product WO3�g� reacts withwater to formWO2�OH�2�g�. Although such reactions can take placein the gas phase, the lack of associated kinetics data renders theirinclusion in the analysis futile. In the present study, all the further

Fig. 1 Baseline nozzle configuration.

Table 4 Nozzle inlet flow conditionsa

Combustion-product species (nonmetallized AP/HTPB)YH2O

0.29YCO2

0.22YCO 0.11YH2

0.01YN2

0.10YHCL 0.27

Motor operating conditionspt, MPa 6.9, 10, 15, 25, 35, 45Tt, K 3000, 3020, 3040, 3050, 3060, 3065Tamb, K 300

aTungsten density is 19:25 g=cm3, throat radius is 0.57 cm, and averagethickness is 4.8 cm.

Table 5 Experimental studies of refractory-metal nozzleerosion

Reference Nozzle material

Johnston et al. [7] W, Mo, refractory compoundsHaugen [8] Mo, W, Mo=Re, W=Re

Fig. 2 Tungsten nozzle erosion rate with different chemical kinetics

(Unal et al. [22], Kalipatrick and Lott [24], andGreene and Finfork [25])usingWO3�g� as the final oxidation product.

Fig. 3 Tungsten nozzle erosion rate using different final oxidation

products and chemical kinetics from Kalipatrick and Lott [24].§Private communication with S. Haugen, 2007.

44 THAKRE AND YANG

calculations for tungsten nozzles are performed by considering thekinetics data of Kalipatrick and Lott [24] with the following surfacereactions:

Reaction R1:

W � 3H2O!WO3 � 3H2

Reaction R2:

W � 3CO2 !WO3 � 3CO

Figure 4 shows the nozzle erosion rates caused by each of theoxidizing species,H2O andCO2. The former proves to bemuchmoredetrimental (about 3 times) than the latter, a situation similar tographite nozzle erosion [12].

The dependence of the erosion rate on the outer boundarycondition of the nozzle material was examined, because thiscondition affects the nozzle surface temperature and associatedchemical reaction rates. Table 6 lists three different outer boundaryconditions considered herein. The baseline adiabatic condition wasrelaxed by allowing convective heat transfer at the outer boundary.The outer heat-transfer coefficient hamb, estimated from standardcorrelations for turbulent flows over a flat plate, falls in the range of100–500 W=m2 � K, depending on the specific configuration of thenozzle assembly and vehicle speed. Figure 5 shows that thecalculated erosion rate decreases slightly with enhanced heat transferat the outer boundary, due to the decrease in the nozzle inner surfacetemperature. The erosion rate calculated by employing the adiabaticouter nozzle boundary thus represents the upper limit. Table 6summarizes the results. All the calculations presented subsequentlyare based on the adiabatic outer boundary of the nozzle materialunless mentioned otherwise.

Figure 6 shows the entire nozzle flowfield in terms of thetemperature, Mach number, and mass fractions of H2O, CO2, CO,andH2. TheMach number increases from0.28 at the inlet to 2.3 at theexit, but the temperature decreases monotonically from a valueslightly less than that of the chamber condition to about 1900 K. Theendothermic surface reactions and conductive heat transfer throughthe nozzlematerial help lower the surface temperature. The thicknessof the species-concentration boundary layer (�c) is greater than itsvelocity counterpart and can be estimated using a simple order-of-magnitude analysis as follows:

�c ��������������Deff�f

p(13)

where Deff is the effective mass diffusivity on the order of10�2 m2=s. Theflow residence time �f is approximated as the ratio ofthe nozzle length to the average axial velocity and has a value ofabout 0.1 ms. Based on Eq. (13), the concentration boundary-layerthickness is about 1 mm, which is close to that observed in Fig. 6.

At a quasi-steady-state condition, whether the species diffusetoward or away from the surface depends on the sign of the speciesgradient dYk=dr at the surface, as governed by the followingequation, which is a rearranged form of Eq. (8):

�d ~Ykdr

���� �_!k � ��g ~Yk ~ur�

��gDkm

(14)

The species gradient in turn depends on two factors: the rate ofproduction/consumption of species _!k and surface blowing velocityur. BecauseH2O andCO2 are consumed at the surface (i.e., negative_!k), their gradients are always positive, with diffusion toward thesurface. The direction of CO and H2 diffusion, however, is affectedby the relative magnitudes of _!k and ur. As a result of the dominanceof reaction R1 in dictating the erosion rate, a large amount of H2 isproduced and diffuses away from the surface. In the case of graphitenozzle erosion [12], because both the surface reactions of C�s� withH2O andCO2 yield CO, the higher value of _!CO dominates the effectof ur. Consequently, the mass fraction of CO at the surface becomesmuch higher than that in the core flow, causing it to diffuse awayfrom the surface throughout the nozzle length. For a tungsten nozzle,however, only a limited amount of CO emerges from the surfacethrough reaction R2, leading to a low value of _!CO. In the upstreamconverging section of the nozzle, as a consequence of high surfaceblowing (higher erosion rate), the mass fraction of CO is lower thanin the core flow and CO diffuses toward the surface. In thedownstream region, however, the low surface blowing (lowererosion rate) reverses the situation. Figure 7 clearly shows thedistributions of mass fraction of CO and temperature at the nozzlesurface.

Figure 8 compares tungsten and graphite nozzle erosion rates forthe same operating condition and nozzle configuration. The erosionrate at the graphite nozzle throat (0:124 mm=s) is about 2.6 times thatof tungsten (0:047 mm=s). This suggests that tungsten(19:25 g=cm3), which is approximately 10 times denser thangraphite (1:9 g=cm3), actually exhibits a higher mass consumptionrate (g=cm2 � s). The lower erosion rate of tungsten is thus attributedto its higher density. It is clear from Eq. (12) that the net nozzleerosion rate is inversely proportional to its material density.

Fig. 4 Tungsten nozzle erosion rate due to individual oxidizing species

H2O and CO2.

Table 6 Effect of outer boundary condition of nozzle on the tungsten erosion ratea

Outer boundary condition @Tc=@r�ro � hamb�Tc;o � Tamb� Erosion rate at throat, mm=s Inner surface temperature at the throat, K

Adiabatic 0.047 2450hamb � 300 W=m2, Tc;o � 600 K 0.042 2390hamb � 500 W=m2, Tc;o � 600 K 0.036 2320

aTamb � 300 K, Tt � 3000 K, and pt � 6:9 MPa.

Fig. 5 Effect of nozzle outer boundary condition on erosion rate.

THAKRE AND YANG 45

The radial distributions of temperature and oxidizing-speciesconcentrations are instrumental in identifying the mechanisms ofnozzle material erosion. Figures 9 and 10 show the radialdistributions of temperature at the throat and at a downstreamlocation, respectively. The temperature at the throat decreasesmonotonically from 2785K at the centerline to 2450K at the surface.The corresponding values at the downstream location are 2280 and2550 K, respectively, but the radial profile does not decreasemonotonically. The temperature rise near the surface results from the

dissipation of the flow kinetic energy into thermal energy. The lowersurface temperature at the throat, as compared with its counterpart atthe downstream location, is attributed to the higher rate ofendothermic heterogeneous reactions at the throat. Figures 11 showthe radial distributions of species mass fractions at the nozzle throat.The finite concentrations ofH2O andCO2 at the surface indicate thaterosion is kinetically controlled.

The effect of chamber pressure on nozzle erosion rate was alsostudied. Figure 12 shows the distribution along the entire length ofthe tungsten nozzle at various chamber pressures. The maximumvalue is attained in the throat region for all cases. It is worth noting

Fig. 6 Distributions of temperature, Mach number and mass fractions ofH2O, CO2, CO, andH2 in the nozzle (Tt � 3000 K and pt � 6:9 MPa, withsurface reactions and conductive wall).

Fig. 7 A tilted view of the nozzle showing distributions of temperatureand mass fractions of CO (Tt � 3000 K and pt � 6:9 MPa, with surface

reactions and conductive wall).

Fig. 8 Comparison of erosion rates of graphite and tungsten nozzles.

Fig. 9 Radial distribution of temperature at the tungsten nozzle throat.

Fig. 10 Radial distribution of temperature at a location downstream of

the nozzle throat.

46 THAKRE AND YANG

from Fig. 2 that for surface reactions with lower activation energies,the peak in the erosion-rate profile spreads more widely in theupstream region than the cases with higher activation energies. Asthe pressure increases, the peak becomes more prominent at thethroat, as evidenced in Fig. 12. The surface temperature, pressure,and activation energies of surface reactions are thus the three factorsmost instrumental in determining the characteristics of the erosion-rate profile. Figure 13 shows the linear variations of the tungsten andrhenium erosion rates at the throat with the chamber pressure.Because the convective heat-transfer rate increases with pressure,there is a corresponding rise in the erosion rate. The lower value ofthe rhenium erosion rate, as compared with that of tungsten, can beattributed to the lack of including CO2 reaction in the former case.The observations based solely on the effect of theH2O reaction showthat the erosion rates of tungsten and rhenium are comparable,especially taking into account the scatter in the kinetics data from theexperimental studies.

To validate the current analysis, calculations were performed tosimulate the nozzle-erosion experiments by Johnston et al. [7] andHaugen [8]. The nozzle geometries and inlet flow conditions exactlysimulate those in the corresponding experimental studies. Table 7summarizes the inlet conditions in terms of mass fractions of thespecies and motor operating conditions. Haugen reported thetungsten nozzle recession data from an earlier subscale motor firingof the Penguin Mk3 missile. The erosion rate was deduced frommeasurements of the thrust and chamber pressure over the firingduration. The nozzle material was tungsten with 2% La2O3, and thepropellant used was nonmetallized AP/HTPB (88:11) with 1%additives. The nozzle surface showed some deposition of silicareleased from the upstream thermal insulation. Figure 14 shows thecalculated nozzle erosion rate along with the nozzle contour for theinlet conditions listed in Table 7. The erosion-rate profile peaks in thenozzle throat region. Because the nozzle contour is very smooth, theerosion profile does not exhibit any irregularity, unlike the situationwith the nozzle contour in Fig. 2. The calculated erosion rate of0:047 mm=s closely matches the measured value of 0:040 mm=s.The slight overprediction may be attributed to two facts: theassumption of an adiabatic outer boundary and the neglect of gas-phase reactions of tungsten oxide with H2O. Silica deposition,detected in the postfiring analysis, may also contribute to thedecrease in themeasured throat diameter. If heat transfer is allowed atthe outer boundary, the predicted erosion rate is lowered marginallyto 0:042 mm=s, as shown in Fig. 14. The uncertainty of the kineticsdata employed for heterogeneous reactions also affects the calculatederosion rate.

The second validation was performed based on the extensiveexperimental work of Johnston et al. [7] to evaluate the performanceof several rocket-nozzle materials, including W, Mo, refractory-metal carbides, and graphite. The instantaneous throat radius wasobtained from measurements of the thrust and chamber pressure, aswell as postfiring analyses. The high-density tungsten nozzlesperformed well with only slight-to-moderate erosion. One of thepropellants used was Arcite 368 (polyvinyl chloride and ammoniumperchlorate). The inlet conditions based on this nonmetallizedpropellant are summarized in Table 7. Although the arc-cast tungstennozzle obtained from the commercial supplier showed an erosionrate of 0:028 mm=s, the molybdenum nozzle insert did not erode at

Fig. 11 Radial distributions of species concentrations at the nozzle

throat.

Fig. 12 Distributions of tungsten erosion rates along the nozzle length

at various chamber pressures.

Fig. 13 Effect of chamber pressure on tungsten and rhenium erosion

rates at the nozzle throat.

Fig. 14 Calculated tungsten nozzle erosion rate for the experimental

study by Haugen (private communication with S. Haugen, 2007).

Table 7 Inlet conditions for simulations of experiments

Reference YCO2YH2O

YH2YCO YHCL YN2

pt, MPa Tt, K

Haugena b 0.21 0.28 0.01 0.10 0.29 0.11 5.6 3000Johnston c[7] 0.20 0.27 0.01 0.12 0.30 0.10 6.9 2860Haugen d[8] 0.16 0.26 0.01 0.20 0.27 0.10 6.9 2810

aPrivate communication with S. Haugen, 2007.bTungsten density is 19:25 g=cm3, throat radius is 0.4 cm, and average thickness is3 cm.cTungsten density is 19:25 g=cm3, molybdenum density is 21:20 g=cm3, throat radiusis 0.39 cm, and average thickness is 1 cm.dTungsten density is 19:25 g=cm3, rhenium density is 21:20 g=cm3, throat radius is0.53 cm, and average thickness is 2 cm.

THAKRE AND YANG 47

all. Figure 15 shows calculated erosion-rate profiles for the tungstenand molybdenum nozzle inserts, along with the nozzle contour. Themaximum convective heat-transfer and erosion rates occurredslightly upstream of the throat. For a curved nozzle with a large angleof convergence, the wall heat transfer usually reaches its maximumjust before the throat. Such a phenomenon has been observed bothexperimentally [35] as well as computationally [11] for curvednozzles. The calculated erosion rate with an adiabatic outer nozzleboundary is 0:041 mm=s for tungsten and nearly zero formolybdenum, as shown in Fig. 15. If this adiabatic condition isrelaxed, then the predicted erosion rate decreases. The measurederosion rate could, in fact, be lower due to the fact that gas-phasereactions between volatile tungsten oxides and H2Omay reduce theoxidation potential of the propellant.

The third validation study was based on experiments by Haugen[8]. The nozzle materials tested on the subscale motor were W, Mo,Mo=Re, and W coated with Re. A reduced-smoke AP/HTPBpropellant was employed with varying compositions. Table 7summarizes the nozzle geometry and inlet conditions. The validity oferosion-rate measurements remains to be clarified [8], as heavy silicadeposition from the upstream region protected the nozzle surfacefrom exposure to the combustion species for a major part of the firingduration. The erosion rate for the tungsten nozzle was reported to bein the range of 0:02–0:03 mm=s, whereas the values for othermaterials were not mentioned. It was concluded, however, that the

best insert material was rhenium-coated tungsten. The molybdenuminsert showed negligible erosion, with the propellant having a lowerflame temperature (2476K), but the erosion increased drastically at ahigher flame temperature (2810 K). This is in contrast to thenegligible erosion for the Mo insert observed by Johnston et al. [7]for a similar flame temperature (2860 K). Figure 16 shows thecalculated erosion rates for tungsten and rhenium, along with thenozzle contour. The erosion rates at the nozzle throat are 0.040 and0:032 mm=s for W and Re, respectively. Table 8 summarizes thecomparison of all the calculated and measured erosion rates at thethroat. The slight discrepancies may be attributed to the uncertaintiesof the chemical kinetics data for surface reactions and the adiabaticouter boundary condition employed.

VI. Conclusions

A comprehensive analysis has been established to predict thechemical erosion of refractory-metal (tungsten, molybdenum, andrhenium) nozzle materials in rocket motors with nonmetallized AP/HTPB solid propellants. The primary focus was on the tungstennozzle. The predicted erosion rates matched reasonably well withthree different sets of measurements. H2O proved to be the moredetrimental oxidizing species than CO2 in dictating tungsten nozzleerosion. The material recession rate is controlled by heterogeneouschemical kinetics and increases linearly with the chamber pressure.To improve the model predictivity, more accurate chemicalproperties for heterogeneous reactions of refractory metals withoxidizing species at realistic rocket-motor operating conditions areneeded. The gas-phase kinetics of volatile tungsten oxides and steamalso needs to be studied further and incorporated into the analysis toaccount for the fact that a reduction inH2O concentrationmay reducethe erosion rate. The erosion rate for tungsten was found to be muchlower than for graphite, but comparable with that of rhenium. Theleast erosion was exhibited by molybdenum, for which theimplementation, however, is restricted to propellants with lowerflame temperatures (less than 3000 K) due to its low melting point.

Appendix A: Thermodynamic Analysis of SurfaceChemical Reactions

The spontaneity and favorability of a chemical reaction depend onthe change in Gibbs free energy associated with the reaction. Gibbsfree energy is defined as follows:

G�H � TS (A1)

The change in Gibbs free energy at a constant temperature andpressure is given by

�G��H � T�S (A2)

A chemical reaction is spontaneous if �G < 0. Equation (A2)indicates that even for endothermic reactions (�H < 0), a situationsimilar to the present surface reactions, the process can bespontaneous if �G < 0: that is, if T�S >�H. Clearly, hightemperature favors the progress of an endothermic reaction. Inaddition, the more negative the value of�G, the more favorable thereaction becomes.

When chemical kinetics data for a given surface reaction are notavailable, the related change in Gibbs free energy can help identifywhether the reaction will take place. It can further assist in the

Table 8 Comparison between calculated and measured nozzle erosion rates

Reference (propellant) Nozzle material _rexpt, mm=s _rmodel, mm=s

Haugena (AP/HTPB) W with 2% La2O3 0.040 0.047 (adiabatic outer boundary) 0.042 (hamb � 300 W=m2 � K)Johnston et al.[7] (Arcite 368) Arc-cast W 0.028 0.041 (adiabatic outer boundary) 0.037 (hamb � 500 W=m2 � K)

Molybdenum Negligible NegligibleHaugen [8] (AP/HTPB) W with 2% La2O3 0.020–0.030 0.040 (adiabatic outer boundary) 0.035(hamb � 500 W=m2 � K)

Recoated W Not mentioned 0.032 (adiabatic outer boundary)

aPrivate communication with S. Haugen, 2007.

Fig. 15 Calculated tungsten and molybdenum nozzle erosion rate for

the experimental study by Johnston et al. [7].

Fig. 16 Calculated tungsten and rhenium nozzle erosion rate for the

experimental study by Haugen [8].

48 THAKRE AND YANG

characterization of final oxidation products when there is anuncertainty and/or difference of opinion associated with it. Table A1lists Gibbs free energies for different surface reactions at 1 atm and2800 K. The latter is typical of the nozzle surface temperature in arocket-motor environment. Because the pressure dependence ofsurface reactions is not known to be significant, the analysis shouldbe reasonably valid at higher pressures. All the thermodynamicproperties are from McBride et al. [36]. No calculation for therhenium-oxidation reaction was performed due to the lack ofthermodynamic data for rhenium oxides. In the case of tungstenoxidation when different final products are considered, the change inGibbs free energy is negative for WO3�g� but positive for WO2�g�,indicating the spontaneity and prevalence of the former product. Thechange in Gibbs free energy of Mo oxidation by CO2 has a morenegative value than its counterpart forW. The former reaction is thusspontaneous and likely to occur at 2800 K.

Acknowledgments

Thisworkwas supported by theU.S. Office ofNaval Research as apart of Multi-University Research Initiative (MURI) project fundedunder contract N00014-04-1-0683, with JudahGoldwasser and CliffBedford as contract monitors. The authors are deeply grateful toSteinar Haugen of Nammo Raufoss AS, Norway, Richard Yetter,and Justin Sabourin for experimental data and technical advice.

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Reaction T, K �HR, kJ �SR, J=K �GR ��HR � T�SR, kJW� 3CO2�g� !WO3�g� � 3CO�g� 2800 498.250 180.866 �17:174W� 2CO2�g� !WO2�g� � 2CO�g� 2800 564.023 184.158 48.380Mo� 3CO2�g� ! MoO3�g� � 3CO�g� 2800 435.629 174.674 �52:425

THAKRE AND YANG 49

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S. SonAssociate Editor

50 THAKRE AND YANG